Back to Mth Roots

As mentioned in §3.4, there are
different numbers
which satisfy
when
is a positive integer.
That is, the
th root of
, which is
written as
, is not unique--there are
of them. How do
we find them all? The answer is to consider complex numbers in
polar form.
By Euler's Identity, which we just proved, any number,
real or complex, can be written in polar form as

where
and
are real numbers.
Since, by Euler's identity,
for every integer
, we also have

Taking the
th root gives

There are
different results obtainable using different values of
, e.g.,
. When
, we get the same thing as
when
. When
, we get the same thing as when
, and so
on, so there are only
distinct cases. Thus, we may define the
th
th-root of
as